P-space

Suppose $X$ is a completely regular topological space. Then $X$ is said to be a P-space if every prime ideal in $C(X)$, the ring of continuous functions on $X$, is maximal.

For example, every space with the discrete topology is a P-space.

Algebraically, a commutative reduced ring $R$ with $1$ such that every prime ideal is maximal is equivalent to any of the following statements:

• $R$ is von-Neumann regular,

• every ideal in $R$ is the intersection of prime ideals,

• every ideal in $R$ is the intersection of maximal ideals,

• every principal ideal is generated by an idempotent.

When $R=C(X)$, then $R$ is commutative reduced with $1$. In addition to the algebraic characterizations of $R$ above, $X$ being a P-space is equivalent to any of the following statements:

• every zero set is open

• if $f,g\in C(X)$, then $(f,g)=(f^2+g^2)$.

Some properties of P-spaces:

1. Every subspace of a P-space is a P-space,

2. Every quotient space of a P-space is a P-space,

3. Every finite product of P-spaces is a P-space,

4. Every P-space has a base of clopen sets.

For more properties of P-spaces, please see the reference below. For proofs of the above properties and equivalent characterizations, see http://planetmath.org/?op=getobj&from=books&id=46here.

Title	P-space
Canonical name	Pspace
Date of creation	2013-03-22 18:53:13
Last modified on	2013-03-22 18:53:13
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	6
Author	CWoo (3771)
Entry type	Definition
Classification	msc 54E18
Classification	msc 16S60